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Mathematical and Computational Applications, Vol. 17, No. 2, pp. 164-175, 2012

VIBRATION ANALYSIS OF ROTOR BLADES OF A FARM WIND-POWER

PLANT

K. Turgut Grsel1, Tufan oban2, Aydogan zdamar3

1 Department of Mechanical Engineering, Faculty of Engineering, Ege University,35100 Bornova Izmir / Turkey, turgut. gursel@ e ge. edu.tr

2Product Development, Schneider Group3 Department of Naval Architecture and Marine Engineering, Naval Architecture and

Maritime Faculty, Yldz Technical University, 34349 Beikta, Turkey,

Abstract- In wind-power plants like in all engineering structures, vibrations are takeninto consideration in design phases in order to avoid resonance. This condition occurs

when the frequency of the exciting force coincides with one of the natural frequenciesof system, which causes dangerously large amplitudes. In the present study, naturalfrequencies of the Rotor blades of NACA (National Advisory Committee forAeronautics) 4415 and NASA/Langley LS(1) 421MOD series of wind-power plants thatdeliver energy to be consumed by a farm family with four persons are calculated.Therefore, after designing the rotors, their natural frequencies are determined first byRayleighs Method and next by finite element method. Further for both rotor blades,resonance analysis is carried out by the found excitation of external forces.

Key Words- Wind-power plant; Propeller blade; Vibration analysis; Resonance; Finiteelement method; Rayleighs Method.

1. INTRODUCTION

In engineering structures, a condition of resonance is encountered, when thefrequency of the exciting force coincides with one of the natural frequencies of thesystem. This can result in dangerously large amplitudes, which leads the system tocollapse. Therefore, in all systems exposed to vibrations and variable forces, thedetermination of natural frequencies is of great interest.

Wind energy, an environment-friendly renewable energy resource that is utilizedin recent years more intensively, is transformed into the mechanical energy in wind-

power plants through the rotor blades ([1], [2]). Nowadays, modern components ofwind-power plants especially rotor blades and towers are constructed in elastic andslender characteristic that conducts these structures and components to inclination forvibration ([3], [4], [5]). Hence, vibrations in such engineering structures have to beinvestigated in design phases.

In this regard, El Chazly [6] carried out static and dynamic analysis of a taperedand twisted blade but with constant chord length and symmetric aerofoil-shaped cross-sections of NACA 0015 series. Maalawi and Negm [7] formulated and appliedappropriate optimisation model for wind turbine design, which strongly was simplified

by modelling symmetric aerofoil-shaped cross-sections as circle ones and by neglectingtwist angle. Younsi et al. [8] investigated dynamical behaviours of a wind turbine blade,

whose twist was neglected and its cross-sections were full, homogenous and made of
mailto:[email protected]:[email protected]

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165K. T. Grsel, T. oban and A. zdamar

wood. Baumgart [9] developed a rod model for wind turbine blades for applications, inwhich a significant reduction in the number of co-ordinates was required. However, the

modelling includes uncertainties so that model shapes do not match well withexperimentally found results. Bechly and Clausen [10] chose a profile similar to a

NACA 4412 series for designing a composite wind turbine blade using finite elementanalysis. The produced model, especially its mesh was simplified.

Nevertheless, due to maximizing energy to be obtained from the plants, theblades of NACA and NASA/LS series used by worldwide known manufacturers of thewind power plants possess the following features: Tapered (variable chord length),twisted and asymmetric aerofoil-shaped cross-sections. Thus, these characteristicsrequire highly developed software programs for modelling and additionally forvibration analyses of blades, which were used in this study from the same reason.Moreover in this research, any assumptions that provide simplifications regarding the

forms of the blades were avoided as far as possible.In wind farms nowadays, wind turbine plants with high nominal powers

operate. Hence a considerable amount of engineering expertise went into design of theseplants. However, less attention was given to wind turbines, which produce low power inkilowatts. These plants usually deliver electrical energy to single use. For this purposein this study, a wind-power plant that will supply energy to be consumed by a farmfamily with four persons was projected, after two optimal wind rotors were designed.Then, vibration analyses of the rotor blades of NACA 4415 and LS 0417 series wereexecuted, because these rotor blades are mostly subjected to vibrations out of thecomponents of wind-power plants.

Afterward, the each fundamental natural frequency of the both rotor blades wascalculated by Rayleighs Method. Further, the natural frequencies and their directionswere determined by the finite element method (FEM). Finally, the frequency of theexciting forces affecting the rotor blades was determined to evaluate possible resonance.

2.1 Design of the rotor blades

2. MATERIAL

In the present study, it was assumed that a person in Turkey consumes annualelectrical energy of 1400 kWh. Thus, the annual electrical energy that is exhausted by afarm family with four persons was calculated to be 5600 kWh. When the annual

electrical energy needed by this family will be obtained only from the wind-powerplant, the radius of the rotor required is found as follows:The power obtained from kinetic energy in air stream is determined by the Eq.

(1) ([4]). 2 3

Here:

Pw

=CP R V2 w

(1)

Cp= 0.40 Power coefficient = 1.23 kg/m3 Density of the airR : Radius of the rotor blade [m]Vw : Wind velocity [m/s]

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i) Number of blades: z = 3,

ii) Radius of rotor (Blade length): R=2 m,iii)iv)

Tip-speed ratio:Profile of the blade:

tip = 7NACA 4415 and LS 0417

r

In this study, the wind velocity was taken as a constant of Vw = 6 m/s that is theannual average wind velocity in wind-rich areas of Turkey ([4], [11], [12], [13]). When

the wind-power affects in interval t, the wind energy is found as:E=P

wt. (2)

The Eqs. (1) and (2) deliver the annual electrical energy that is obtained from thewind-power plant within a year by neglecting site wind characteristics, capacity andavailability factor. This energy value to be obtained matches the maximal use and/orloading of the wind power plants without considering any cost analyses:

E =CR

2V

38760 . (3)

By means of the Eq (3),

year P2

w

5 600 000 [Wh] =0.401.23R2

638760 ,

2the radius of the rotor blades (of the wind-power plant) is found asR=1.96 m, and it waschosen as R = 2 m, whereas Weibull wind distribution was neglected primarily due tofocusing on the modal analysis of the blades to be determined. The blade number wasdetermined asz = 3, because wind-power plants withz =3 deliver generally the highestlevel of electrical energy ([3], [4], [5]).

In the design phase, the angle of attack was chosen as D = 10 as wellrecommended in ([4]); the optimal twist angle was calculated by the Eq. (4):

2 R = arctan=

. (4)

twist3

D tip

In Eq. (4), r denotes the distance between the rotor hub and the chosen cross-section. Consequently, the profiles of the blade cross-sections were drawn for each r.Finally, the parameters in the rotor design of the wind-power plant were determined asfollows ([4]):

topt

(r) = 1 163 1.3

rsin213

arctan10

7r

(5)

The Eq. (5) delivers the optimal chord length of the rotor blades for each cross-section r ([4]). Further, in order to design the rotor blades, blade length (radius of therotor) was divided into 11 cross-sections of the same length. The first cross-section

begins at r = 0.05 R and the eleventh completes at r = R.

As an example, the optimal chord length of the profile at r = 0.05.

2 = 0.1 m wascalculated from the Eq. (5) as follows, and this calculation was repeated for each cross-section so that Table 1 was filled out:

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t (0.1) =1 16

0.1sin21

arctan10

= 0.2966 [m] .

opt 3 1.3 3 7 0.1

But the chord length at r = 0.1 R in Table 1 was chosen as t = 0.2233 m to avoidconstructive problems while assembling the blades at the rotor hub.

2.2 Modelling of the rotor blades

In the present study, the standard profiles of NACA 4415 and LS 0417 serieswere chosen for both rotor blades, and the coordinates of these profiles were given inTable 2.

Table 1. Optimal chord lengths calculated from the standard profiles of NACA 4415and LS 0417 series

NACA4415

r

[m] 0.1 0.29 0.48 0.67 0.86 1.05 1.24 1.43 1.62 1.81 2

topt[m] 0.2233 0.2662 0.2124 0.1679 0.1367 0.1147 0.0985 0.0862 0.0765 0.0688 0.0624

LS

0417

r

[m] 0.1 0.29 0.48 0.67 0.86 1.05 1.24 1.43 1.62 1.81 2topt[m] 0.2233 0.2662 0.2124 0.1679 0.1367 0.1147 0.0985 0.0862 0.0765 0.0688 0.0624

Table 2. Coordinate ratios of cross-section profiles of NACA 4415 and LS 0417 series

NACA4415

x/t 0 0.25 0.5 0.75 1

yt/t 0 0.115 0.1 0.07 0yb/t 0 0.035 0.02 0.015 0

LS 0417x/t 0 0.25 0.5 0.75 1

yt/t 0 0.115 0.09 0.065 0yb/t 0 0.065 0.055 0.03 0

Table 3. Coordinates of outer cross-section profiles of NACA 4415 and LS 0417 seriesfor r = 0.48 m

NACA4415

Outer

topt=212.4 mmx -53.1 0 53.1 106.2 159.3

yt

0 24.426 21.24 14.868 0yb 0 -7.434 -4.248 -3.186 0

LS 0417Outer

topt=212.4 mmx -42.5 0 42.5 85.0 127.5

yt 0 19.55 15.3 11.05 0

yb 0 -11.05 -9.35 -5.1 0

Table 4. Coordinates of inner cross-section profiles of NACA 4415 and LS 0417 seriesfor r = 0.48 m

NACA4415

Inner

topt=176.6 mm

twist=120

x -44.039 0 44.04 88.08 132

yt 0 20.258 17.616 12.331 0

yb 0 -6.166 -3.523 -2.643 0

LS 0417

Inner

topt=176.6 mm

twist=120

x -42.5 0 42.5 85.0 127.5

yt 0 19.55 15.3 11.05 0

yb 0 -11.05 -9.35 -5.1 0

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The blades were modelled by filling out the solid volumes between the elevencross-sections ([13]). Therefore first, the coordinates of the cross-sections were

calculated. As an example, only the calculations for the cross-section r = 0.48 m of theblades were given in Table 3 and 4 and all cross-sections of both were shown in Fig. 1aand Fig. 1b.

Each modelled profile cross-section describes a closed curve that consists of 8points. As seen in Fig. 2a and Fig. 2b, to model the volume of the each blade, twoguidelines that run at the beginning point of the curves were drawn. Further by means ofthese guidelines and the eleven cross-sections, the whole volume of the each blade wasmodelled as given in Fig. 3a and Fig. 3b.

Figure 1a. Perspective view of the eleven cross-sections of NACA 4415; Figure 1b.Perspective view of the cross-sections of LS 0417 series

Figure 2a. Jointing the blade cross-sections with guidelines for NACA 4415 series;Figure 2b. Jointing the blade cross-sections with guidelines for LS 0417 series

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2

Figure 3a. Solid model of the rotor blade accomplished by cross-sections of NACA4415 series

Figure 3b. Solid model of the rotor blade accomplished by cross-sections of LS 0417series

3. METHOD

3.1 Vibration analysis by Rayleighs method

Since the blades twisting along the blade axes (r) are slight, it was assumed thatthe change of initial moment of the blades in direction of the blade axes should be

relatively insignificant. In addition, the wind power plants are machines with very lowangular speeds (20-120 rpm). At very high wind speeds they must be stopped. Thereforethe characteristics of the both blades were investigated by means of the Rayleighsmethod in the case of stopping period of the blades and in the case of its rotating periodwith very low angular speeds. According to Rayleighs Method, the natural frequencyof the each blade was calculated with the Eq. (6), which is certainly the upper limit of it.

2x1 d2 q EI(x) dx 2

dx

2 = k =

x0

mx1

(6)

q 2(x)dx

x0

In the Eq. (6), Eand I(x) denote the modulus of elasticity in N/m2 and the moment ofinertia in m4, respectively. In the present study, weldable aluminium alloy AlMg5 wasused as material for both rotor blades. Hence, the modulus of elasticity of this material

was determined asE=71109 [N/m2] .In determining cross-deflection of the blade by means of the value of q in Eq.

(6), the expression q= (x/R)2 is differentiated twice according to x, which is a variablefor the radius of rotor or the blade length. Subsequently, the following constant isobtained:

q2 = 2

=constR

2

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4

x

9

T

Finally, the following term, in which is the density of the alloy AlMg5 given as =2700 kg/m3, denotes;

(x) =A(x) .When arranging the Eq. (6), the equation of the natural frequency is obtained as

follows:

2 =

constE

x1

x1

I(x)dxx0

x A(x) R dx 0

The integral terms in Eq. (6) were calculated by MATLAB program and the results ofNACA 4415 series given below were obtained.

200200 4

x I(x)dx= 1.762 107 [m

5] ; A(x)

dx =9.155105 [m3]

10 10 R By means of these found values, the first natural frequencies of both rotor blades of

NACA 4415 and LS 0417 series were determined as follows:

2

=const7110 1.76210

7 =15531.66 [rad2/s

2]

NACA

2700 9.155105

NACA

=124.63

[rad/s] ; fNACA

=19.836

[Hz]

LS =111.62 [rad/s] ; fLS =17.765 [Hz]

3.2 Vibration analysis by finite element package ANSYS

Vibration caused by the excitation of external forces is called forced vibration.The vibration is continued through periodical excitations. The amplitude of thevibration depends on system parameters and exciter characteristics. The Eq. (8) givesthe general differential equation of a forced vibration:

[M]{u}+[D]{u}+[K]{u}={F(t)}

(8)

In the Eq. (8), M, D, and Kdenote the matrices of coefficients in FEM as follows ([14],

[15], [16], [17]):

M =M(m) = (m)

H(

m)

H(m)

dV(m) ; M: Mass matrix of vibrating system,

m m V(m)

T

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D =D(m) = (m)H(m)

H(m)

dV(m) ; D: Matrix of viscous damping coefficients,

m m V(m)

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T

T

T

K=K(m) = B(m)

C(m)

B(m)

dV(m)

; K: Matrix of elastic spring

coefficients,m m V(m)

(m): Density of the mass of the mth element

(m): Viscous damping coefficients

H(m): Matrix of extension

B(m): Matrix of deformation

C(m): Matrix of elasticity of the mth element

F (t) = RB + RS+RC ; External forces

RB

=

RB

m

(m) =

H(

m)

m V

fB(m)

dV(m) ; RB: Forces of volume

(m)

RS

=RSm

(m) = HS(m)

m S(m)

fS(m)

dS(m) ; RS: Forces of surface

RC = F0 : Single forcesu: Displacement.

Free vibration begins with any excitation that delivers energy to the system, and

then, this excitation disappears. When neglecting the friction and damping, the vibrationis theoretically continued to infinity through a steady transformation of the potentialenergy into the kinetic energy and vice versa due to the displacement and elasticdeformation. Indeed while transforming the mechanical energy, free vibrations decreasewith time and they damp down finally due to losses caused by frictions.

Decrease of amplitudes depends on parameters of the system. The excitationdetermines the boundary conditions of the motion; hence, it affects amplitudes of thevibrations. Consequently, the main characteristic of this vibration is a function of

physical properties of the system. In absence of viscous damping, the equation of freevibrations yields as follows:

[M]{u}+[K]{u}={0}

(9)

When arranging the Eq. (9) according to harmonic motion, the Eq. (10) isobtained:

([K]2 [M]){u}={0}

(10)i i

In Eq. (10), w2i denotes eigenvalues, and i shows degree of freedom, and the terms {ui}mean eigenvectors. The square root of the eigenvalue delivers the natural frequencieswi. The eigenvectors of {ui} express the modes. Mode extraction in ANSYS as a sub-

program calculates eigenvalues and eigenvectors ([17]).The lowest potential and deformation energy causes the first mode that explains

the natural frequency. The second and third modes need more energy; consequently,they possess more deformation energy. Generally, many natural frequencies have to becalculated in engineering structures that are exposed to harmonic forces. This obtainedinformation is used in dynamic analyses.

The modes, like in natural frequencies, depend on the weight and rigidity ofstructures and on distribution of the mass. Additionally, the aerodynamic elasticity ofthe rotor blades of wind-power plants has significant influence on vibrations and mode

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shapes of them. The mass moment of inertia can reflect the combined effect of all these

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parameter. While increasing the inertia moment of mass, the natural frequencydecreases due to the inverse ratio.

The aim of the modal analysis is to determine the natural frequencies of a systemin order to avoid resonance so that the natural frequencies do not take place in thefrequency range of excitations of the system. In this context, the FEM that is used formodal analysis of complicated engineering structures is very suitable and sometimes theonly method.

In the commercial software ANSYS, different methods for calculating vibrationmodes are available. In the present study, the method of Block Lanczos was chosenfor the following reasons, although it requires great data banks ([17]): Satisfactory results in analyses of complicated models including solid elements,

shell elements and beams are obtained. It is suitable for modes, whose boundary conditions are not given. It is favourable for analyses of modes that have to be calculated in a defined

interval.In this study, the characteristics of the both blades also were investigated by

means of the developed soft ware based on the FEM in the case of stopping period ofthe blades and in the case of its rotating period with very low angular speeds, asexplained in section 3.1. In order to execute modal analysis within the ANSYS

program, first, the blades were modelled and meshed (Fig. 4a and Fig. 4b). Then, typeof the analysis was chosen; the properties of the material and the boundary conditionswere applied. Subsequently, calculations were executed. The stresses that occur due tovibrations were determined and visualised in Fig. 5. So, necessary vibration modes and

directions of rotations were obtained as given in Table 5.

Figure 4a. Meshed model of the rotor blade of NACA 4415 series; Figure 4b. Meshedmodel of the rotor blade of LS 0417 series

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Figure 5. Visualising of stress distribution of NACA 4415 series

4. RESULTS OF ANALYSIS AND CONCLUSION

Aerodynamic and elastic forces occurring cause aero-elastic vibrations, and thesevibrations arise around all three axes. First, rotor blades generally vibrate across to the

plane, in which the rotor rotates (1st

mode, Table 5). The rotor blades clap into the winddirection and vice versa (clap vibration) ([4]). Further, rotor blades vibrate in the plane,in which the rotor rotates (2nd mode, Table 5). A typical vibration form of a rotor withthree blades is such that two blades simultaneously in rotation direction and one bladein other direction vibrate (sway vibration) ([4]). These both types of vibrations in 3 rd,4th, 5th modes, and these vibrations and additionally torsional ones around all three axesin higher modes occur simultaneously (Table 5).

The kinetic energy in the wind stream is transformed into the mechanical energyin the rotor shaft through the rotor blades. In this study, it was assumed that thegenerator, which is driven by the rotor shaft, is a synchronous generator. Hence, therotational speed of the rotor shaft remains as constant at all events. As determined in

section design of rotor blades, the following parameters were used in the Eq. (10):Vw = 6 m/s ; R = 2 m ; tip = 7 for the profiles of NACA 4415 and LS 0417 series.By the Eq. (10), the angular velocity or rotational speed of the rotor blade was obtainedas:

= 21 rad/s or n = 200.6 rpm.

Vptip =

Vw

=RV

w

(10)

The 1st and 2nd modes of the natural frequencies of the both rotor blades ariseoutside of the frequency range of the exiting force such as n1 NACA= 1188.4 rpm ; n2NACA = 2711.9 rpm and n1 LS= 1059.0 rpm; n2 LS= 2374.7 rpm so that any danger of

resonance does not exist for the both rotor as given in Fig. 6.

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LS-1

w

[rad/s]

300

250

200

150

NACA = 124.45 [rd/s]LS = 110.89 [rd/s]

NACA-4415

LS-0417

100

50

0

0 1000 2000 3000

= 21 [rd/s] Workingfrequency

n [rpm ] Resonancezones

seriesFigure 6. Resonance zones of the rotor blade of NACA 4415 and LS 0417

Table 5. Values of modal vibrations and rotational directions of NACA 4415 and LS0417 series

NACA 4415 LS 0417

Mode f [Hz] [rad/s] f [Hz] [rad/s] Type of vibration Direction

1 19.806 124.45 17.649 110.89 Bending Aroundx axis

2 45.197 283.99 39.578 248.68 Bending Aroundy axis

3 80.061 503.04 69.774 438.40 Bending Aroundx andy axis



6 299.74 1883.4 269.74 1694.83 Torsional bendingBending: Aroundx andy axis

Torsion: Aroundzaxis



8 520.33 3269.4 462.7 2907.23 Torsional bending Bending: Aroundx andy axisTorsion: Aroundzaxis

9 539.82 33918 599.18 3764.76 Torsional bendingBending: Aroundx andy axis




In this study, two rotor blades for a wind-power plant that produces electricalenergy for a farm family with four persons were designed. The vibration analyses of the

blades from NACA 4415 and LS 0417 series were executed first by Rayleighs Methodand next by the FEM. By the both methods, the 1 st modes of free vibrations of the

blades were calculated as 1 NACA 124.5 rad/s and 1 LS111.0 rad/s. The highermodes of the natural frequencies and the rotational directions for both blades were

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determined by the FEM. The frequency of the forced vibration was found to be fF =3.343 Hzthat is very distant from possible resonance zones. Thus, it cannot be expected

under these operating conditions that the blades and/or rotors do not get into resonance.

5. REFERENCES

1. A. Ozdamar, K.T. Gursel, G. Orer, Y. Pekbey, Investigation of the potential of wind-waves as a renewable energy resource: by the example of Cesme-Turkey. Renewableand Sustainable Energy Review. 8: 581-592, 2004.2. A. Ozdamar, N. Ozbalta, A. Akin, E.D. Yildirim, An application of combined windand solar energy system in Izmir. Renewable and Sustainable Energy Review. 9: 624-637, 2005.3. D. A. Spera, Wind Turbine Technology: Fundamental Concepts of Wind Turbine

Engineering. 2nd ed., ASME Press: New York, 1995.4. E. Hau, Wind Power Plants. Springer Verlag, 2. Auflage, 1996. (In German)5. R. Gasch, Wind Power Plants. B.G. Teubner Stuttgart, 3. Auflage, 1996. (In German)6. N. M. El-Chazly, Static and dynamic analysis of wind turbine blades using finiteelements method, Comp. Struc. 48(2): 273-290, 1993.7. K. Y. Maalawi, H. M. Negm, Optimal frequency design of wind turbine blades. J.Wind Eng. & Ind. Aerod. 90: 961-986, 2002.8. R. Younsi, I. El-Batanony, J. B. Tritsch, H. Naji, B. Landjerit, Dynamic study of awind turbine blade with horizontal axis.Eur. J. Mech. A/ Solids. 20: 241-252, 2001.9. A. Baumgart, A mathematical model for wind turbine blades. Journal of Sound andVibration. 251(1): 1-12, 2002.10. M. E. Bechly, P. D. Clausen, Structural design of a composite wind turbine bladeusing finite element analysis. Comp. Struc. 63(3): 639-646, 1997.11. A. Hepbasli, A. Ozdamar, N. Ozalp, Present status and potential of renewableenergy sources in Turkey.Energy Sources. 23: 631-648, 2001.12. A. Ozdamar, H. Yildiz, O. Sar, Wind energy utilization in a house in Izmir- Turkey.Int. J. of Energy Res. 25: 253-261, 2001.13. T. Coban, Vibration analysis of a rotor blade of a wind-power plant. DiplomaThesis, Ege University, Izmir. (In Turkish), 2003.14. J. Li, S. T. Lie, Z. Cen, Numerical analysis of dynamic behavior of stream turbine

blade group.Finite Elements in Analysis und Design. 35: 337-348, 2000.

15. J. K. Bathe,Finite Element Methods. Springer-Verlag. (In German), 1986.16. M. H. Hansen, Aeroelastic stability analysis of wind turbines using an eigenvalueapproach. Wind Energy. 7:133-143 (DOI: 10.1002/we.116), 2004.17. ANSYS, Users Manual, USA, 2005.

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